On the adaptive wavelet estimation of a multidimensional regression function under $\alpha$-mixing dependence: Beyond the boundedness assumption on the noise

Christophe Chesneau (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We investigate the wavelet estimation of a multidimensional regression function $f$ from $n$ observations of a $\alpha$-mixing process $(Y,X)$, where $Y=f(X)+\xi$, $X$ represents the design and $\xi$ denotes the noise. In most papers considering this problem either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of $f$ in its construction) or it is supposed that the noise is bounded (excluding the Gaussian case). In this paper we go far beyond this classical framework. Under no boundedness assumption on the noise, we construct two kinds of adaptive term-by-term thresholding wavelet estimators enjoying powerful mean integrated squared error (MISE) properties. More precisely, we prove that they achieve "sharp" rates of convergence under the MISE over a wide class of functions $f$. "Sharp" in the sense that they coincide with the optimal rate of convergence in the standard {\it i.i.d.} case up to extra logarithmic terms.

Mots clés: Nonparametric regression; $\alpha$-mixing dependence; Adaptive estimation; Wavelet methods; Rates of convergence

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Christophe Chesneau. On the adaptive wavelet estimation of a multidimensional regression function under $\alpha$-mixing dependence: Beyond the boundedness assumption on the noise. 2012. ⟨hal-00727150v2⟩